A distributional approach to 2D Volterra dislocations at the continuum scale

Abstract

We develop a theory to represent dislocations and disclinations in single crystals at the continuum (or mesoscopic) scale by directly modelling the defect densities as concentrated effects governed by the distribution theory. The displacement and rotation multi-valuedness is resolved by introducing the intrinsic and single-valued Frank and Burgers tensors from the distributional gradients of the strain field. Our approach provides a new understanding of the theory of line defects as developed by Kröner [10] and other authors [6, 9]. The fundamental identity relating the incompatibility tensor to the Frank and Burgers vectors (and which is a cornerstone of the theory of dislocations in single crystals) is proved in the 2D case under appropriate assumptions on the strain and strain curl growth in the vicinity of the assumed isolated defect lines. In general, our theory provides a rigorous framework for the treatment of crystal line defects at the mesoscopic scale and a basis to strengthen the theory of homogenisation from mesoscopic to macroscopic scale.